1. Introduction
To offset anthropogenic climate change, the world is reducing its reliance on fossil fuels with renewable sources of energy. Among these sources is offshore wind energy, whose global capacity is expected to grow from 56 GW recorded at the end of 2021 to 380 GW by 2030 (Global Wind Energy Council 2022). Offshore wind energy is vital to the plan set by the United Kingdom to achieve carbon neutrality by 2050 (Department for Business, Energy & Industrial Strategy 2020), as it accounted for 11% of the United Kingdom’s gross energy production in 2020 (Office for National Statistics 2021). Furthermore, the United Kingdom plans to raise the current offshore wind capacity from 12.7 to 50 GW by 2030 (U.K. Wind Energy Database 2022; Department for Business, Energy & Industrial Strategy 2022). By 2050, Germany intends to achieve an installed offshore capacity of 50–70 GW (Agora Energiewende et al. 2020) as a part of the European Union efforts to reach 450-GW offshore wind capacity, with the North Sea contributing over 84% (Wind Europe 2019). Outside Europe, the United States will deploy 30 GW worth of offshore wind turbines by 2030 (National Renewable Energy Laboratory 2021).
To maximize generated power per occupied surface area and per support structure, the diameter and hub height of wind turbines are increasing to reach higher altitudes with stronger, more constant winds (e.g., Fleming and Probert 1984). However, Antonini and Caldeira (2021) found that, beyond a threshold size, wind farms with taller turbines shed longer and more intense wakes while experiencing a substantial drop in power density (i.e., generated power per occupied surface area). These wakes risk higher turbulence on downwind turbines, leading to higher structural fatigue (Pryor et al. 2021). The wakes also reduce the available wind resource for downwind farms, limiting their performance (e.g., Lundquist et al. 2019) and possibly leading to legal disputes (van der Horst and Vermeylen 2010). Moreover, Akhtar et al. (2021) argued that poor planning of offshore wind farms can limit future exploitation of wind resources. As a consequence, it is key to understand the impact of wind turbines on the surrounding environment. This impact, for modern-day offshore wind farms, can extend to tens of kilometers downwind, leading to interactions at a mesoscale rather than at a microscale for small wind farms (Platis et al. 2018).
A challenge in the numerical modeling of wind farms in mesoscale frameworks is the horizontal grid spacing, which ranges from 1–2 km for relatively small domains to tens of kilometers for global simulations. At such grid sizes, wind turbines cannot be explicitly resolved, forcing modelers to parameterize wind-turbine effects on the atmospheric flow. One early approach was to represent the thrust exerted by wind turbines through increasing the surface roughness length of the grid cells that contain wind turbines (Keith et al. 2004; Barrie and Kirk-Davidoff 2010). However, modern-day wind turbines can impact a substantial fraction of the atmospheric boundary layer (ABL) due to their larger diameters and hub heights extending above the surface layer. In this case, increased surface-roughness approaches fail to resolve the elevated impacts of the rotor, leading to inaccurate results (Fitch et al. 2013).
Multiple wind-farm parameterizations have been proposed by simplifying Eq. (1) to get an approximate tendency of the horizontal wind speed ∂U/∂t (Fischereit et al. 2022). The simplification of Eq. (1) is done differently for each parameterization as discussed in appendixes A and B. Version 4 of the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2019, p. 89) includes the parameterization by Fitch et al. (2012, hereafter Fitch) as a standard option within the Mellor–Yamada–Nakanishi–Niino (MYNN) boundary layer scheme (Nakanishi and Niino 2009). Other parameterizations include the explicit wake parameterization of Volker et al. (2015, hereafter Volker), Abkar and Porté-Agel (2015, hereafter Abkar), Pan and Archer (2018, hereafter Pan), and Redfern et al. (2019, hereafter Redfern).
Previous studies, excluding the studies introducing the parameterizations themselves, mostly compared the performance of Fitch’s and Volker’s parameterizations. A series of studies focusing on onshore wind farms in Iowa compared Fitch’s and Volker’s parameterizations using WRF v3.8.1 (Pryor et al. 2019, 2020; Shepherd et al. 2020). The studies shared the conclusion that power generation was higher using Volker’s parameterization, and Fitch’s parameterization resulted in stronger wind speed deficits that extended for longer distances downwind. Pryor et al. (2019) highlighted that the differences in power generation from Fitch’s and Volker’s parameterizations were significant for wind-farm economic considerations, with differences in generation up to 10% of the wind farms’ rated power. However, they could not conclude which parameterization was better. Also, Pryor et al. (2020) showed that turbulence levels over the Iowa wind farms using Volker’s parameterization were low, and were much less than that of Fitch’s parameterization. However, Archer et al. (2020) showed that WRF versions up to v4.2.1 suffered from a bug that impacted the advection of turbine-induced turbulence, and they argued that the conclusions in the studies that used a bug-affected version of WRF need to be handled with caution.
Using WRF v4.2.2 that is not affected by the bug, Pryor et al. (2022) compared Fitch’s parameterization to Volker’s parameterization, focusing on lease areas off the East Coast of the United States through sets of 5-day-long simulations using 1922 IEA 15-MW reference wind turbines (Gaertner et al. 2020). Similar conclusions to the Iowa studies were reported for these offshore wind farms. Over the North Sea, Larsén and Fischereit (2021a) compared simulated wind speed and turbulent kinetic energy (TKE) from Fitch’s and Volker’s parameterizations to airborne measurements by Platis et al. (2018) and Siedersleben et al. (2020). They concluded that Volker’s parameterization underestimated TKE over a wind farm and predicted higher wind speeds than Fitch’s parameterization. To the best of our knowledge, only Porté-Agel et al. (2020) went beyond these two parameterizations and briefly compared the vertical profiles of turbine-induced thrust and TKE resulting from the parameterizations of Fitch, Abkar, and Blahak (Blahak et al. 2010) to large-eddy simulations (LES) based on the results of Abkar and Porté-Agel (2015). They found that the results of Abkar’s parameterization were in better agreement with LES than the others, whereas Fitch’s parameterization overestimated turbine-induced turbulence.
Owing to the scarcity, shortness, and/or spatial locality of available field measurements, there is no preferred wind-farm parameterization in the literature (Fischereit et al. 2022). Hence, the purpose of the present study is to provide an in-depth comparison between published wind-farm parameterizations by extending previous studies beyond the parameterizations of Fitch and Volker. We consider the parameterizations of Fitch, Volker, Abkar, Pan, and Redfern through a case study of multiple offshore wind farms located in the North Sea. The impact of different parameterizations on wind speed, wind direction, TKE, and temperature is assessed relative to published measurements, and wake effects and power generation are also analyzed, with a particular focus on interactions between wind farms operating in close proximity.
The rest of this study is organized as follows. The setup of the numerical experiments is discussed in section 2. Section 3 compares near-surface wind speed to satellite imagery and discusses the horizontal extent of wind farms’ wakes. Section 4 compares wind speed and TKE to airborne measurements, whereas section 5 shows vertical profiles of multiple flow variables and highlights the role of wind turbines in changing these variables. Section 6 focuses on the impact of wake on power generation, and the sensitivity of power generation to some tunable parameters within each wind-farm parameterization. The results of sections 3–6 are further discussed in section 7 on a broader scale to draw conclusions regarding the considered wind-farm parameterizations, which are then summarized in section 8. The historical evolution of wind-farm parameterizations is discussed in detail in appendix A, and the equations of turbine-induced thrust and turbulence for the considered wind-farm parameterizations are listed in appendix B. In appendix C, we discuss some of the differences in turbulence generation among the considered wind-farm parameterizations. The impact of applying the energy correction term [Eq. (B4)] on turbine-induced thrust and turbulence is shown in appendix D. Finally, appendix E justifies not using the deviation of wind speed signal recorded by the FINO-1 meteorological mast in the calculation of TKE.
2. Problem definition and diagnostics
To evaluate the differences of the considered wind-farm parameterizations, multiple WRF simulations are performed, whose details are examined in section 2a, and whose results are presented in sections 3–6. Section 2b introduces the definition of some diagnostic variables that will be used throughout these sections.
a. Case study details
In 2017, three field measurement expeditions were performed over the North Sea using a crewed aircraft (a Dornier 128-6 operated by TU Braunschweig) to investigate the effect of offshore wind farms on the local atmosphere (Platis et al. 2018; Siedersleben et al. 2018a,b, 2020). The expeditions were done on 9 August, 14 October, and 15 October 2017, and were denoted cases I, II, and III, respectively. Siedersleben et al. (2020) reported that they were able to correctly simulate the flow field for case II (Bärfuss et al. 2019) only, which was performed over the Gode Wind 1 and 2 wind farms (hereafter, Gode Wind farms), and consisted of six transect flights and six vertical profiles with their start and end times described in Larsén and Fischereit (2021a). Siedersleben et al. (2020) used WRF v3.8.1 with the built-in Fitch parameterization to represent the offshore wind turbines. One of their suggestions for future work was to resimulate case II using Volker’s parameterization, which was done later by Larsén and Fischereit (2021a), who identified some of the differences between Fitch’s and Volker’s parameterizations, and investigated the low-level jets that appeared during this time period. Here, we further analyze case II using the parameterizations of Fitch, Volker, Abkar, Redfern, and Pan, which are described in appendix B, while exploring the sensitivity of each wind-farm parameterization to its own parameters. Another goal of this study is to quantify the influence of wakes generated from upwind farms on the power generation of downwind neighboring farms.
Table 1 shows the details of the wind farms that were operating in October 2017 and are considered in this study. The locations of these wind farms are shown in Fig. 1, with the turbines of each farm colored differently. The coordinates of the individual wind turbines were obtained from the Sentinel-1 project imagery (European Space Agency 2018) and are available in Ali et al. (2022). Other wind farms (Butendiek, DanTysk, Sandbank, Nordergründe, and Horns Rev 1 and 2) are not included as they are located more than 100 km transverse to the dominant flow direction.
Description of the wind farms considered in this study with the year of commission listed. Wind turbines’ performance curves are available in Larsén and Fischereit (2021b) and in Ali et al. (2022), whereas the commission dates were extracted from 4C Offshore (2022).
The boundaries of the three nested domains used in WRF are colored in red and have grid sizes of 15, 5, and 1.67 km. A zoom-in on the location of the wind farms included in this study is shown. Domain boundaries are curved as they are defined using a Lambert projection in WRF, but are plotted on a background generated using a latitude–longitude projection. The green dotted lines over the Gode Wind farms are the transect paths of the aircraft.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
The boundaries of the three nested domains used in WRF are colored in red and have grid sizes of 15, 5, and 1.67 km. A zoom-in on the location of the wind farms included in this study is shown. Domain boundaries are curved as they are defined using a Lambert projection in WRF, but are plotted on a background generated using a latitude–longitude projection. The green dotted lines over the Gode Wind farms are the transect paths of the aircraft.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
The boundaries of the three nested domains used in WRF are colored in red and have grid sizes of 15, 5, and 1.67 km. A zoom-in on the location of the wind farms included in this study is shown. Domain boundaries are curved as they are defined using a Lambert projection in WRF, but are plotted on a background generated using a latitude–longitude projection. The green dotted lines over the Gode Wind farms are the transect paths of the aircraft.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
We use WRF v4.3.3 to simulate case II using three nested domains with horizontal grid sizes of 15, 5, and 1.67 km, respectively, following the setup of Siedersleben et al. (2020). The upper boundary of the simulation domain is the 10-hPa pressure level. We use 80 vertical levels with 17 levels below the height of 200 m, and 8–12 levels intersecting a wind turbine’s rotor depending on its diameter and hub height. The initial and boundary conditions were extracted from the ERA5 reanalysis (Hersbach et al. 2020). The WRF double-moment 6-class microphysics scheme (Lim and Hong 2010), the Kain–Fritsch cumulus parameterization scheme implemented on the 15-km domain only (Kain and Fritsch 1993; Kain 2004), the RRTMG shortwave and long-wave radiation scheme (Iacono et al. 2008), the Mellor–Yamada–Nakanishi–Niino level-2.5 planetary boundary layer scheme (Nakanishi and Niino 2009), and the Noah land surface model (Niu et al. 2011) were chosen for the numerical setup. The TKE advection option was turned off following Siedersleben et al. (2020) who found that, for the atmospheric conditions of case II, turning on the TKE advection option led to an underestimation of TKE and wind speed deficit over the Gode wind farms, as well as an overestimation of TKE in the wake compared to airborne measurements. Because we are regenerating case II in the present study, we followed their setup while acknowledging that this may not be optimum for other scenarios. All simulations start at 0000 UTC 14 October 2017, with 12 h of spinup period.
Table 2 lists the details of the conducted numerical experiments. For Fitch’s parameterization, we investigate the effect of the turbulence correction factor [α in Eq. (B8)] by conducting two experiments, for which α = 1 (F100) and α = 0.25 (F25) as suggested by Archer et al. (2020). The F100-NE experiment is performed to determine the effect of turning off the energy correction [Eq. (B4)] when α = 1. As shown in Eq. (B11), Volker’s parameterization depends on the parameter
A summary of the numerical experiments conducted in this study. The name of the base experiment for each wind-farm parameterization is written in bold.
b. Definition of datasets and diagnostics
Before exploring the results of the experiments listed in Table 2, datasets and some assisting diagnostic variables are introduced. These diagnostics are defined in terms of the horizontal and vertical impacts of the wind farms.
1) Synthetic aperture radar
Section 3 shows a comparison between simulated wind speed and a processed satellite image observed by the synthetic aperture radar (SAR). SAR measures small-scale ocean surface roughness, allowing wave height to be related to the wind field through application of a geophysical model function (James 2017). The comparison is done for simulated 10-m horizontal wind speed against that derived from a processed SAR image on the 1717 UTC 14 October 2017 overpass on the Sentinel-1 satellite. The processed image has a grid spacing of order 100 m, which is much lower than the model grid size we used. However, the empirical relations used to infer the 10-m wind speed from surface measurements assume a neutrally stable ABL. Hence, for a stable ABL, as in the current case study, the SAR data are expected to underestimate the 10-m wind speed, making the comparison to the SAR image more qualitative than quantitative.
2) Wake extent and length
3) Transect flights over the Gode wind farms
Section 4 verifies the conducted experiments against airborne measurements above the Gode Wind farms. The measurements were taken between 1420 and 1608 UTC 14 October 2017, at an approximately constant height of 250 m above sea level, along the paths indicated by green dashed lines in Fig. 1 (Bärfuss et al. 2019). The aircraft sampled the flow field at a rate of 100 Hz, and its recordings were averaged using a moving-average filter with a window size of 2 km (Platis et al. 2018).
4) FINO-1 meteorological mast
Section 5 presents a comparison of the FINO-1 mast measurements of wind speed and wind direction against the conducted simulations. The FINO-1 mast was commissioned in 2003 over the North Sea (54°00′53.5″N, 6°35′15.5″E) with a height of approximately 101 m above sea level, to collect measurements of wind speed, wind direction, and temperature among other measurements for research purposes (Muñoz-Esparza et al. 2012; FINO1 2022).
5) Calculating wind direction
6) Budget terms of
7) Power generation
Section 6 discusses wind-farm power generation which can be impacted by two sources of wakes: interarray wakes and intra-array wakes. Interarray wakes are the wakes shed by upwind farms, whereas intra-array wakes are the wakes shed by upwind turbines within the same wind farm. By definition, the NT experiment does not include these wake effects, and the power calculated within this experiment represents the available power for generation if each wind turbine operates independently from the other turbines (hereafter, ideal power). The other experiments account for these wake effects in power calculation (hereafter, actual power). The generated power is written in the form of a capacity factor, which is a wind farm’s power divided by its rated power as listed in Table 1.
3. Verification against synthetic aperture radar data
In this section, near-surface simulated wind speed is compared to SAR observations, and wakes shed by the wind farms are investigated.
Figure 2a shows the 10-m wind speed from the SAR, whereas Figs. 2b–l are output from the experiments in Table 2. The F100-NE experiment is not shown in Fig. 2 as its wind speed field is close to that of F100, and the role of the energy correction is discussed in section 5 and appendix D. Figure 2 indicates that simulated wind speeds in some experiments (e.g., F100, R100) are qualitatively comparable to the satellite image except for some small-scale structures that the simulations could not reproduce (e.g., the thin streaks in Fig. 2a). Moreover, the simulated wakes are wider than those indicated by the SAR, because the used wind-farm parameterizations evenly distribute the turbine-induced thrust over the grid cell containing a wind turbine, rather than resolving the wake of each individual turbine. Hence, using relatively coarser grid sizes leads to wider and shorter wakes than those observed from field measurements. Figure 2 also shows that wind turbines impact near-surface wind speed for tens of kilometers downwind even though hub heights are 80–110 m above sea level.
Horizontal wind speed at 10 m above sea level at 1717 UTC 14 Oct 2017 (colored). Satellite image is processed from the data of DTU (2022). The black contours indicate the outline of the wind farms and are combined for simplicity when wind farms are close to each other.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Horizontal wind speed at 10 m above sea level at 1717 UTC 14 Oct 2017 (colored). Satellite image is processed from the data of DTU (2022). The black contours indicate the outline of the wind farms and are combined for simplicity when wind farms are close to each other.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Horizontal wind speed at 10 m above sea level at 1717 UTC 14 Oct 2017 (colored). Satellite image is processed from the data of DTU (2022). The black contours indicate the outline of the wind farms and are combined for simplicity when wind farms are close to each other.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Equation (13) suggests that a rise in turbulence (q in the first term of the right-hand side) enhances the turbulent momentum flux in the vertical direction from higher altitudes downward (
The near-surface wind speed acceleration in Fig. 2 is local to a wind-farm’s footprint and is followed by a reduction in wind speed along the farm’s wake. To examine this wind speed deficit, Fig. 3 shows horizontal planes of normalized wind speed deficit [Eq. (2)] at a height of 90 m above sea level at 1717 UTC 14 October 2017. Normalized wind speed deficit values below 2% were blanked, because it is common to have perturbations of 2% or less between different numerical experiments due to the chaos in the atmosphere. Hence, only normalized wind speed deficit values above 2% are included because they are more clearly related to the existence of the wind farms.
Normalized wind speed deficit (λ), 90 m above sea level at 1717 UTC 14 Oct 2017 (colored). The wind farms’ footprint is colored in black. All normalized wind speed deficits above 20% are included in the 15%–20% range.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Normalized wind speed deficit (λ), 90 m above sea level at 1717 UTC 14 Oct 2017 (colored). The wind farms’ footprint is colored in black. All normalized wind speed deficits above 20% are included in the 15%–20% range.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Normalized wind speed deficit (λ), 90 m above sea level at 1717 UTC 14 Oct 2017 (colored). The wind farms’ footprint is colored in black. All normalized wind speed deficits above 20% are included in the 15%–20% range.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Figure 3 shows that the experiments of Volker’s parameterization predicted shorter and thinner wakes than the other parameterizations due to lower values of turbine-induced thrust (see Fig. 7a later in the article). Within the experiments of Volker’s parameterization, Figs. 3d–f indicate that smaller values of the quantity
There is a high variation in the prediction of wake length between different wind-farm parameterizations (Fig. 3). To quantify these variations, Fig. 4 shows the time average of the normalized wake extent [Eq. (4)] and wake length [Eq. (3)] of the Gode Wind farms, 90 m above sea level, whereas Tables 3 and 4 show the percentage differences in normalized wake extent and wake length, respectively, between different wind-farm parameterizations. Wind-farm wake extent increases with turbine-induced thrust as more momentum is extracted from the flow. However, turbine-induced turbulence enhances the mixing process, leading to faster recovery and relatively shorter wakes. Therefore, a wind-farm’s wake extent results from a balance between these two effects. For instance, F25 and R25 have approximately the same values of turbine-induced thrust as F100 and R100, but with considerably less turbulence (Table 2). This makes the length of the λ = 20% level of the Gode Wind farms 25.6% and 28.2% longer in F25 and R25 than in F100 and R100, respectively, due to the reduced turbulence-induced mixing (Table 4). Moreover, turbulence in V100 is considerably less than that of F100 (see Fig. 7d later in the article), which is in favor of longer wake extent for V100. However, the turbine-induced thrust of F100 is larger, overtaking the effect of lower turbulence in V100, and making V100 predict a λ = 20% level for the Gode Wind farms that is 42.7% shorter than that of F100 (Table 4).
(top) Normalized wake extent (NWE) of the Gode Wind farms. (bottom) Wake length (
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
(top) Normalized wake extent (NWE) of the Gode Wind farms. (bottom) Wake length (
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
(top) Normalized wake extent (NWE) of the Gode Wind farms. (bottom) Wake length (
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Percentage difference in normalized wake extent (NWE) of the Gode Wind farms as per the results shown in Fig. 4 for different normalized wind speed deficit (λ) ranges. For experiment i and reference experiment j, the percentage difference is calculated as (NWEi − NWEj)/NWEj × 100%.
Turning off the energy correction in Fitch’s parameterization (F100-NE) led to increases in normalized wake extent (Table 3) and wake length (Table 4) for all wind speed deficit ranges compared to F100. For instance, the λ = 20% level for F100-NE is 4% larger than that of F100. The correction function applied in Pan’s parameterization had a substantial impact on wake extent, as PAN predicted a λ = 20% level that is 11.3% shorter than that of F100 (Table 4), whereas the horizontal surface area with wakes of λ ≥ 20% was 46.5% less than F100 (Table 3). Within Volker’s parameterization, varying
4. Verification against research aircraft flights over the Gode wind farms
Instead of satellite imagery, here we compare simulated wind speed and TKE to airborne measurements over the Gode wind farms.
Figure 5 shows the profiles of wind speed and TKE from WRF along the paths of the transect flight segments above the Gode Wind farms (green dashed lines in Fig. 1). For some of the flight segments, the simulated wind speed in many of the experiments was approximately 1–2 m s−1 higher than the observations (e.g., Figs. 5c,g). To compare observed and simulated wind speed deficits, measurements of wind speed over the Gode wind farms without wind turbines are required. Such measurements are not available. We have instead used a surrogate for the wind speeds that the aircraft would have measured if the Gode wind farms did not exist by assuming a linear wind speed profile between the two ends of the wind farm (the shaded regions in Fig. 5). The choice of a linear profile is motivated by the NT experiment which has no turbines included and shows an almost linear wind speed variation between the ends of the Gode Wind farms (Fig. 5).
Wind speed and TKE variation along the aircraft’s flight path over the Gode Wind farms. The shaded regions indicate the spatial range of averaging simulated and observed wind speed deficits. Each row represents a single flight segment.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Wind speed and TKE variation along the aircraft’s flight path over the Gode Wind farms. The shaded regions indicate the spatial range of averaging simulated and observed wind speed deficits. Each row represents a single flight segment.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Wind speed and TKE variation along the aircraft’s flight path over the Gode Wind farms. The shaded regions indicate the spatial range of averaging simulated and observed wind speed deficits. Each row represents a single flight segment.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
By averaging the simulated wind speed profiles in Fig. 5 between 53.95° and 54.15° latitude (shaded regions in Fig. 5) for all flights, there was an averaged wind speed deficit of 0.9 m s−1, whereas its observed counterpart is 0.84 m s−1. Hence, the conducted experiments predicted acceptable values of wind speed deficit over the Gode Wind farms, and the consistent bias between the simulations and observation was probably attributed to the background flow. However, the variation in simulated wind speed deficit between different wind-farm parameterizations reached 1.7 m s−1, such as that between F100 and V120 (Fig. 5i). In general, Volker’s experiments underestimated wind speed deficit over the Gode Wind farms, whereas the experiments of Fitch’s and Redfern’s parameterizations along with A100 and PAN were closer to the observed wind speed deficit (Fig. 5).
Observed wind speed over the Gode Wind farms exhibited a sudden rise at the southern edge of the farms slightly below 54° latitude (e.g., Fig. 5i). However, our simulations, similar to the simulations of Siedersleben et al. (2020) and Larsén and Fischereit (2021a), could not capture this behavior. Siedersleben et al. (2020) attributed this behavior to the stable stratification of the ABL, forcing the upcoming flow to deflect more to the sides of the Gode Wind farms than above them. No sudden increase in wind speed was observed at the northern edge of the Gode Wind farms due to the orientation of the Gode Wind farms relative to the upcoming flow from the southwest, making the aircraft path at the northern edge in a location of less horizontal wind shear. In contrast, there is an acute turn of the flow at the southern edge leading to larger horizontal wind shear. Siedersleben et al. (2020) added that the extent of this large horizontal wind shear region was approximately 2 km, which cannot be resolved with the grid sizes of a mesoscale model (i.e., larger than 1 km), because the scale of the resolved phenomena should be a few multiples of the grid size (e.g., 4–7 multiples).
For TKE profiles in Fig. 5, the absence of an explicit source of turbulence in Volker’s parameterization resulted in low magnitudes of turbulence above the Gode Wind farms, reaching a maximum value of approximately 0.3 m2 s−2, whereas observed TKE exceeded 1 m2 s−2 for most of the Gode Wind farms’ footprint (e.g., Fig. 5f). These low TKE magnitudes were close to that of the NT experiment, which showed an approximately flat TKE profile. The other parameterizations (i.e., Fitch, Abkar, Pan, and Redfern) include a source of turbulence and predicted closer TKE values to those of the observations, with F100 and R100 being the closest in most of the flight segments. A spike in TKE of about 2 m2 s−2, and almost 3 m2 s−2 in flight segment 5 (Fig. 5j), is observed from the airborne measurements at the southern edge of the Gode Wind farms accompanying the large horizontal wind shear. WRF simulations could not capture this phenomenon, as well (Siedersleben et al. 2020).
5. Verification against FINO-1 meteorological mast data
Having compared the conducted experiments to SAR (section 3) and to airborne measurements over the Gode Wind farms (section 4), the conducted experiments (Table 2) are compared to the recordings of the FINO-1 meteorological mast. Figure 6 shows temporal profiles of simulated wind speed, wind direction, temperature, and TKE, which are compared to their measured counterparts by the FINO-1 mast, whereas Fig. 7 provides time-averaged vertical profiles of the same variables at the FINO-1 location.
Comparing WRF results to the FINO-1 meteorological mast data. (a),(b) Wind speed at a height of 102 m; (c),(d) wind direction at a height of 91 m; (e),(f) temperature at a height of 101 m; and (g),(h) TKE at the same height of the wind speed. (right) Deviations from the NT experiment averaged during the 12-h period starting at 1200 UTC 14 Oct 2017. The wind speed measurement in (a) is provided as an averaged value (black dots) along with the standard deviation (shaded blue region) and the minimum and maximum values (shaded gray region) during the averaging window. The same definitions apply to wind direction in (c), but with no minimum and maximum values recorded. The convention for wind direction is that east is 0°, and positive rotation is counterclockwise.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Comparing WRF results to the FINO-1 meteorological mast data. (a),(b) Wind speed at a height of 102 m; (c),(d) wind direction at a height of 91 m; (e),(f) temperature at a height of 101 m; and (g),(h) TKE at the same height of the wind speed. (right) Deviations from the NT experiment averaged during the 12-h period starting at 1200 UTC 14 Oct 2017. The wind speed measurement in (a) is provided as an averaged value (black dots) along with the standard deviation (shaded blue region) and the minimum and maximum values (shaded gray region) during the averaging window. The same definitions apply to wind direction in (c), but with no minimum and maximum values recorded. The convention for wind direction is that east is 0°, and positive rotation is counterclockwise.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Comparing WRF results to the FINO-1 meteorological mast data. (a),(b) Wind speed at a height of 102 m; (c),(d) wind direction at a height of 91 m; (e),(f) temperature at a height of 101 m; and (g),(h) TKE at the same height of the wind speed. (right) Deviations from the NT experiment averaged during the 12-h period starting at 1200 UTC 14 Oct 2017. The wind speed measurement in (a) is provided as an averaged value (black dots) along with the standard deviation (shaded blue region) and the minimum and maximum values (shaded gray region) during the averaging window. The same definitions apply to wind direction in (c), but with no minimum and maximum values recorded. The convention for wind direction is that east is 0°, and positive rotation is counterclockwise.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Vertical profiles at the FINO-1 met mast location of (a) turbine-induced thrust [
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Vertical profiles at the FINO-1 met mast location of (a) turbine-induced thrust [
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Vertical profiles at the FINO-1 met mast location of (a) turbine-induced thrust [
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
a. Wind speed
Figure 6a shows that the simulated wind speeds were within the range of measured wind speed during the 24 h of simulation, indicating closeness between simulations and measurements. Between different wind-farm parameterizations, simulated wind speed varied within 0.64–1.41 m s−1, with a 24-h time average of 1 m s−1 (Fig. 6a), showing that wind-farm parameterizations can substantially impact the accuracy of wind speed prediction. Volker’s parameterization predicted the least wind speed deficit with values ranging within 0.5–1.5 m s−1, whereas F25 and R25 predicted the largest wind speed deficits ranging within 1.5–2.5 m s−1 (Figs. 6b and 7c). The small values of wind speed deficit in Volker’s parameterization is due to the small values of turbine-induced thrust compared to the other parameterizations (Fig. 7a). The turbine-induced thrust profiles of Volker’s parameterization show that reducing the quantity
The FINO-1 mast is installed at the upwind section of the Alpha Ventus wind farm, making the blockage corrections (ψ) of Pan’s parameterization near unity at the FINO-1 location due to the absence of blocking upwind turbines. This made the wind speed profiles of Pan’s parameterization close to that of Fitch’s parameterization at the FINO-1 location (Figs. 6b and 7c). Even though Pan’s turbine-induced thrust reduces to that of Fitch’s when the correction function ψ = 1, different turbulence levels were generated [Eqs. (B8) and (B15)], changing turbulence-induced mixing, and the usage of the corrected hub-height wind speed (
b. Wind direction
When compared to the mast measurements, the simulated wind direction profiles [Eq. (5)] of all the considered wind-farm parameterizations had a time-averaged bias of about −5.4° (clockwise) during the last 12 h of simulation (Fig. 6c), which is slightly larger than the standard deviation in measured wind direction, which had a time average of 4.3° during the same period.
Volker’s parameterization was the only one to predict a counterclockwise deflection of wind near hub height at the FINO-1 location (Fig. 6d). The vertical profiles of wind direction of Volker’s parameterization show a counterclockwise deflection at all heights, whereas all the other parameterizations predicted a clockwise deflection from the surface up until slightly above hub height (Fig. 7e). However, LES results showed that the wake of a wind farm in Earth’s Northern Hemisphere near hub height deflects clockwise about its free stream direction as seen from the top (Lu and Porté-Agel 2011; Englberger et al. 2020). This behavior of Volker’s parameterization is primarily related to turbulence magnitudes, which create a southward tendency in the meridional momentum budget of the flow at low altitudes for a wind farm in the Northern Hemisphere (van der Laan and Sørensen 2017). Hence, when turbulence was low, such as in Volker’s experiments (Fig. 7d), there was an underestimation of the southward forcing in the meridional momentum budget, leading to a northward meridional momentum tendency. Analogously, F25 and R25 showed less clockwise deflection near hub height than the other experiments due to their lower turbulence magnitudes (Fig. 7d).
c. Temperature
The time series of temperature at the FINO-1 location (Fig. 6e) indicates a good agreement between simulated and measured profiles, as the average temperature profile among the conducted experiments (except NT) had a time-averaged bias of only 0.07°C compared to measurements. All experiments predicted a slight rise in temperature at the height of measurement due to wind turbines, except for Volker’s experiments which predicted a slight temperature decrease (Fig. 6f). Moreover, the vertical profiles of temperature deviation from the NT experiment indicate a decrease in temperature at all altitudes for Volker’s parameterization (Fig. 7f), whereas all other parameterizations predicted an increase in temperature up to the top-tip of the rotor blades due to stable atmospheric stratification (Baidya Roy and Traiteur 2010). This behavior is related to the lack of turbulence-induced mixing in the experiments with Volker’s parameterization, similar to F25 and R25, which had less turbulence than the others and exhibited a near-zero temperature change near the surface (Fig. 7f).
d. Turbulent kinetic energy
The FINO-1 mast data do not measure TKE explicitly, but include the standard deviation of the time series of the horizontal wind speed magnitude. We show in appendix E that this deviation cannot be used as an approximation of TKE. Hence, TKE will be compared to the NT experiment to quantify the added TKE by each wind-farm parameterization. Figure 6h indicates that the differences in TKE between F100, R100, A100, A90, A80, and PAN are small compared to that with F25, R25, and Volker’s experiments. The reduction in turbine-induced turbulence in F25 and R25 led to a time-averaged 37% reduction in TKE at the mast location, 102 m above sea level (Figs. 6g,h). As for Volker’s experiments, the absence of an explicit TKE source made the turbulence magnitude close to that of the NT experiment (Fig. 6h), similar to the behavior indicated by the airborne measurements over the Gode Wind farms (Fig. 5). Moreover, the vertical profiles of TKE indicate that turbulence magnitudes of Volker’s experiments at the FINO-1 location and at heights below hub height were less than the NT experiment, suggesting turbine-induced turbulence dissipation rather than generation (Fig. 7d).
To explain this, Fig. 8 shows vertical averages of the time-averaged q2 budget components [Eqs. (8)–(12)] along four vertical segments defined by the characteristic heights: sea level, the bottom tip of the rotor, the top tip of the rotor, one diameter above the hub height, and 400 m above sea level. Because turbulence production due to wind shear (
Stacked bars of vertically averaged q2 budget terms of the grid cell containing the FINO-1 mast. The first 400 m above sea level are divided into four segments based on the hub height and diameter of the M5000-116 wind turbine. The segments are (a) beneath the rotor, (b) within the vertical extent of the rotor, (c) above the rotor and up to one diameter above hub height, and (d) up to 400 m. Quantities are time averaged during the 12-h period starting at 1200 UTC 14 Oct 2017. Plotted quantities represent the vertically averaged changes in q2 due to vertical transport
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Stacked bars of vertically averaged q2 budget terms of the grid cell containing the FINO-1 mast. The first 400 m above sea level are divided into four segments based on the hub height and diameter of the M5000-116 wind turbine. The segments are (a) beneath the rotor, (b) within the vertical extent of the rotor, (c) above the rotor and up to one diameter above hub height, and (d) up to 400 m. Quantities are time averaged during the 12-h period starting at 1200 UTC 14 Oct 2017. Plotted quantities represent the vertically averaged changes in q2 due to vertical transport
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Stacked bars of vertically averaged q2 budget terms of the grid cell containing the FINO-1 mast. The first 400 m above sea level are divided into four segments based on the hub height and diameter of the M5000-116 wind turbine. The segments are (a) beneath the rotor, (b) within the vertical extent of the rotor, (c) above the rotor and up to one diameter above hub height, and (d) up to 400 m. Quantities are time averaged during the 12-h period starting at 1200 UTC 14 Oct 2017. Plotted quantities represent the vertically averaged changes in q2 due to vertical transport
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Variation of the vertical gradient of (a) wind direction and (b),(c) the quantity
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Variation of the vertical gradient of (a) wind direction and (b),(c) the quantity
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Variation of the vertical gradient of (a) wind direction and (b),(c) the quantity
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Beneath the rotor (Fig. 8a), turbulence generation in all experiments was dominated by shear production, which was highest in the NT experiment due to the largest wind speed vertical gradients (Fig. 9c), resulting in the largest net turbulence generation within this height range. Within the rotor’s vertical extent (Fig. 8b), turbulence generation was dominated by turbine-induced turbulence production, except for Volker’s experiments. Also, shear production of turbulence for Volker’s experiments was close to that of the NT experiment due to minimal differences in wind speed vertical gradients (Fig. 9b), resulting in a net-change of turbulence that was close to when wind turbines were not modeled (NT). This, along with less turbulence generation beneath the rotor, was responsible for the reduced magnitudes of turbulence relative to NT. Therefore, the absence of a turbine-induced turbulence source term in a wind-farm parameterization can lead to the nonphysical result of wind turbines dissipating turbulence rather than generating turbulence.
Turbulence magnitudes did not have large differences between the other experiments (Figs. 6h and 7b,d), with the exception of F25 and R25, which deviated substantially from the other experiments, especially within the rotor’s vertical extent (Fig. 8b). They also exhibited different behavior regarding the vertical transport of turbulence between one radius and one diameter above hub height (Fig. 8c), as they predicted an upward outflow of turbulence rather than an inflow of turbulence from beneath as observed for the other experiments (excluding Volker’s and NT experiments). This behavior outlines the need to include turbine-induced turbulence in a mesoscale model to obtain accurate simulations. This behavior diminished one diameter above hub height, regardless of the wind-farm parameterization, as shear production and vertical transport were balanced by buoyancy and dissipation (Fig. 8d).
6. Power generation
One of the aims of a mesoscale simulation of wind farms is the prediction of power generation, which is impacted by the wakes of upwind farms, as is discussed in this section. We also discuss the variability of power generation between the alternative wind-farm parameterizations.
To quantify losses of power due to interarray and intra-array wake effects, the difference between the ideal power of a wind farm and its actual power generation for the considered wind-farm parameterizations is examined in Fig. 10, which shows the capacity factors of some of the considered wind farms. The last 12 h of simulation were divided into two 6-h periods: afternoon (red bars) and evening–night (blue bars), because wind speed values in these two periods were different (e.g., Fig. 6a). During the afternoon, wind speeds were close to or exceeded the rated wind speeds (wind speed at which a wind turbine reaches maximum power generation) of the wind-turbine types used in the current simulations (12–16 m s−1), whereas during the evening–night wind speed decreased (e.g., Fig. 6a), causing a reduction in available wind power with magnitudes that were local to each wind farm. For instance, the time-averaged ideal capacity factor of the Gode Wind farms during the evening–night was approximately 0.9, whereas the lowest capacity factor during the same period was that of the Bard Offshore 1 wind farm at 0.6, even though both wind farms had a unity capacity factor during the afternoon.
Capacity factor bars extending from the minimum to the maximum in a wind farm per experiment. The average capacity factor of a wind farm is shown as an open circle. The horizontal dashed lines denote the wind farm–averaged capacity factor of the NT experiment. Red bars correspond to a time average during the 6-h period starting at 1200 UTC 14 Oct 2017, whereas the blue bars are averaged during the 6-h period starting at 1800 UTC 14 Oct 2017. When bars intersect, their intersection extent is shown by a dotted line and colored caps.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Capacity factor bars extending from the minimum to the maximum in a wind farm per experiment. The average capacity factor of a wind farm is shown as an open circle. The horizontal dashed lines denote the wind farm–averaged capacity factor of the NT experiment. Red bars correspond to a time average during the 6-h period starting at 1200 UTC 14 Oct 2017, whereas the blue bars are averaged during the 6-h period starting at 1800 UTC 14 Oct 2017. When bars intersect, their intersection extent is shown by a dotted line and colored caps.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Capacity factor bars extending from the minimum to the maximum in a wind farm per experiment. The average capacity factor of a wind farm is shown as an open circle. The horizontal dashed lines denote the wind farm–averaged capacity factor of the NT experiment. Red bars correspond to a time average during the 6-h period starting at 1200 UTC 14 Oct 2017, whereas the blue bars are averaged during the 6-h period starting at 1800 UTC 14 Oct 2017. When bars intersect, their intersection extent is shown by a dotted line and colored caps.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
In real scenarios, wind farms produce less power than the ideal power due to interarray and intra-array wake effects. To highlight these wake effects, we consider the Bard Offshore 1 wind farm which operated in the near wake of the Veja Mate wind farm (Fig. 1). During the afternoon (red bars), the difference between the ideal (horizontal line) and the actual (open circle) capacity factors of the Bard Offshore 1 wind farm varied between 7.8% (A80) and 27.5% (F25) of the wind farm’s rated power, whereas it varied between 33.7% (V120) and 59% (F25) during the evening–night (Fig. 10). The increase in the difference between ideal and actual power generation during the evening–night is due to the large gradient of the wind turbine’s power curve with respect to wind speed which, for the M5000-116 wind turbine used in the Bard Offshore 1 wind farm, exceeds 800 kW (m s−1)−1 for wind speeds in the range of 9–12 m s−1. Hence, for 80 turbines installed in the Bard Offshore 1 wind farm and an average wind speed deficit of 2 m s−1 during the evening–night, power is reduced by 32% (128 MW) of the wind farm’s rated power due to the combined effect of both wake sources. To isolate the effect of intra-array wakes, we consider the Veja Mate wind farm, which operated within undisturbed wind. During the afternoon, the difference between the ideal and the actual capacity factors was within 0.8%–4.2% (except PAN which predicted 8%) of the wind farm’s rated power versus 7.8%–27.5% for the Bard Offshore 1 wind farm during the same time period. This large difference between two wind farms of similar numbers of wind turbines and close-by locations indicates the impact of interarray wakes, which need to be considered when planning a wind farm’s location.
Within each wind-farm parameterization, power generation can be impacted by varying the parameterization’s tunable constants. Table 5 shows the relative difference of power generation due to different wind-farm parameterizations and due to the tunable constants within each parameterization. The percentages in Table 5 were obtained by averaging the results of the wind farms in Fig. 10. The reduction of turbulence generation in F25 and R25 altered hub-height wind speed (Fig. 7c), resulting in stronger wind speed deficits due to the lack of turbulence-induced mixing. This made F25 predict 3.3% less power generation than F100 during the afternoon and 4.4% less during the evening–night (Table 5). Redfern’s experiments exhibited a similar behavior but with a slightly less reduction in power generation. Switching off the energy correction in Fitch’s parameterization did not have a substantial effect on power generation, as the power generation of F100-NE was only 0.6% less than that of F100. The change in hub-height wind speed due to the correction function of Pan’s parameterization resulted in a substantial drop in predicted power compared to Fitch’s parameterization, because the correction function is cubed in the power calculation equation [Eq. (B19)]. For instance, the power generation in PAN was 6% and 7.5% less than that of F100 during the afternoon and evening–night, respectively (Table 5).
Percentage difference in power generation. For experiment i and reference experiment j, the percentage difference is calculated as (Pi − Pj)/Pj × 100%. Values are averaged across the wind farms of Fig. 10.
For Abkar’s parameterization, the value of ζ affected wind speed substantially (Fig. 7c), causing A80 and A90 to generate 7% and 4.6% more power than A100 during the afternoon of 14 October 2017, respectively (Table 5). During the evening–night, these ratios increased to 20.6% and 11.7%, raising the uncertainty in power generation depending on the adopted value of ζ (Table 5). Volker’s parameterization exhibited less sensitivity to varying
7. Discussion
The purpose of this study was to quantify the key impacts of different wind-farm parameterizations on prediction of wind speed, wind direction, temperature, turbulence, wake extent, and power generation, by referencing each parameterization to an experiment with no wind turbines (NT) and to field measurements. One of the crucial conclusions was the role of turbine-induced turbulence. The airborne measurements indicated an increase in turbulence over a wind farm, whereas WRF simulations indicated that this increase vanished one diameter above a wind turbine’s hub height (Fig. 8). Nevertheless, the absence of an explicit source of turbulence in a wind-farm parameterization can lead to inaccurate predictions of near-surface wind speed (Fig. 2), near-surface temperature (Fig. 7f), and wind direction (Figs. 6f and 7e), as well as the nonphysical result of wind turbines dissipating turbulence near hub height (Figs. 7d and 8). Also, when the turbulence generation source was reduced (F25 and R25), the length of the wake with high wind speed deficit increased (Table 4), and the dynamics of vertical turbulence transport changed (Fig. 8), leading to an underestimation of the near-surface wind speed compared to the satellite image (Fig. 2). Therefore, it is necessary to include an explicit source of turbulence when modeling wind turbines in mesoscale models.
As for the magnitude of turbine-induced turbulence, there is a large difference between Fitch’s formulation, which is based on wind-turbine performance tables, and the formulation of Abkar’s parameterization, which relies on an analytical approximation of the TKE caused by turbine-induced thrust (appendix C). However, the analysis conducted here was insufficient to recommend the better of the two turbulence sources, because the two base experiments F100 and A100 performed well against the airborne measurements over the Gode Wind farms and against the FINO-1 mast data, with F100 performing slightly better against the airborne measurements.
Turbine-induced turbulence also affected the extent of wind farm wake, which was shown to have a substantial impact on power generation. An example of the importance of wake effects on power generation is the Veja Mate wind farm, which was commissioned in May 2017 and is located just upwind of the Bard Offshore 1 wind farm, which had been operative since August 2013. To isolate the impact of Veja Mate on Bard Offshore 1, we conducted a simulation with the base Fitch’s parameterization (F100) including only the wind turbines of Bard Offshore 1. In this hypothetical scenario and during the afternoon of 14 October 2017 when the undisturbed wind speed was close to rated values, the Bard Offshore 1 wind farm had a time-averaged capacity factor of 0.96 instead of 0.82 when the Veja Mate wind farm was included. Removing the Veja Mate wind farm changed the wind speed throughout the Bard Offshore 1 wind farm, and hence changed the intra-array wake effects, as well. So, the difference in power generation was not purely due to interarray wake effects. Nevertheless, the 56-MW difference in power generation was a consequence of the Veja Mate wind farm being constructed directly upwind of the Bard Offshore 1. Hence, interarray and intra-array wake effects cannot be neglected on a system-level planning of a wind farm’s location, size, and architecture.
All the parameterizations considered in this study, with the exception of Pan’s parameterization, neglect wake effects within the same grid cell. Nevertheless, in PAN, the time-averaged and farm-averaged correction function (ψ) was approximately 0.948 and was impacting 30%–56% (average is 43%) of the turbines of a wind farm. These corrections resulted in mild differences in wind speed (Figs. 4 and 7c), wind direction (Fig. 7e), and temperature (Fig. 7f) compared to Fitch’s parameterization, but caused substantial differences in power generation (Fig. 10, Table 5). However, care must be taken when using a parameterization that models subgrid wake effects, because small-enough horizontal grid sizes can resolve some of the modeled subgrid wake effects if the subgrid wake model is not defined relative to the grid size. Resolving some of the modeled subgrid wake effects can occur with Pan’s parameterization as its correction function is not explicitly dependent on grid size, but is a function of wind turbines’ layout within a wind farm. Hence, Pan’s parameterization is best suited to the case of an entire wind farm located in the same grid cell, but at the expense of added computational cost for the calculation of the blockage ratios of wind turbines within each time frame, because they rely on local wind direction.
Overall, wind speed and TKE magnitudes predicted by Fitch’s parameterization were close to the measured values. This result does not necessarily contradict other studies that found excessive turbulence generation using Fitch’s parameterization, because the current comparison is local to the FINO-1 mast and the transect flights over the Gode Wind farms. However, setting α = 0.25 was not optimal in the conducted experiments. In fact, a constant correction factor is unlikely to be suitable for all scenarios. Conversely, Volker’s parameterization underestimated turbulence, resulting in nonphysical predictions in some experiments. Abkar’s parameterization performed well against the airborne and mast measurements, but showed high sensitivity to the value of ζ. A plausible range for ζ would be 0.9–1, because the A80 experiment resulted in a substantial underestimation of wind speed deficit. Theoretically, the value of ζ can surpass unity (Abkar and Porté-Agel 2015), but this poses the possibility of having a negative turbine-induced turbulence, which mathematically cannot occur as long as 0 < ζ ≤ (1 − a)−1, with a being the axial induction factor (appendix B). The same possibility can occur with Fitch’s parameterization if wind turbine tables incorrectly have CT < CP for some wind speeds. The formulation of turbulence generation in Pan’s parameterization does not permit a negative result as long as CT is positive, a benefit that Pan’s parameterization has over the other parameterizations.
Finally, the modifications to Fitch’s parameterization introduced by Redfern et al. (2019) did not have substantial impact on the flow because wind-veer effects were small (Fig. 9a). For instance, wind direction varied approximately linearly within the vertical extent of the rotor (Fig. 9a), with magnitudes between −6.2° in NT and −3.3° in F100 (Fig. 7e). Hence, the wind-veer correction cosθ [Eq. (B21)] was over 0.998 in all of the conducted experiments [i.e., cos(6.2°/2), because hub height is the datum for wind-veer correction], resulting in minimal impact on the calculation of turbine-induced thrust and turbulence. Moreover, the wind speed profile within the rotor’s vertical extent was approximately linear (Fig. 7c), making the hub-height wind speed a good approximation to the rotor-averaged wind speed [Eq. (B20)] if the grid’s vertical levels are assumed to divide the rotor into segments of approximately equal areas. Therefore, the introduced corrections in Redfern’s parameterization can be more pronounced if wind veer is strong and/or the wind speed profile within the rotor’s vertical extent deviated substantially from the linear profile (e.g., due to low-level jets occurring within a rotor’s vertical extent).
8. Summary
Five wind-farm parameterizations (Fitch et al. 2012; Volker et al. 2015; Abkar and Porté-Agel 2015; Pan and Archer 2018; Redfern et al. 2019) were compared using WRF v4.3.3 through a case study on 14 October 2017 of offshore wind farms located in the North Sea. The conclusions presented here are not necessarily representative of all possible scenarios as they were made over a short period of time (12 h) under stable atmospheric conditions. Other periods and stability conditions may lead to different conclusions. The conducted simulations were verified against a satellite image, airborne measurements over the Gode Wind farms, and the FINO-1 meteorological mast data. Fitch’s and Redfern’s wind-farm parameterizations predicted the closest wind speed deficits and turbulence levels to the airborne measurements and the closest wind speeds to the FINO-1 mast data, whereas Volker’s parameterization underestimated both fields. The latter resulted in symmetric turbine-induced thrust and turbulence profiles about hub height, whereas the usage of the energy correction, which is implemented in WRF for Fitch’s parameterization, resulted in an upward shift in the induced profiles. The simulations also indicated that Abkar’s parameterization exhibited high sensitivity to the value of its tunable parameter ζ. The corrections of Pan’s parameterization resulted in a substantial difference in power generation compared to Fitch’s parameterization, while having mild differences regarding wind speed, wind direction, and temperature. Finally, Redfern’s parameterization was close to that of Fitch for most of the experiments as the proposed corrections did not have a substantial impact on turbine-induced thrust and turbulence.
The simulations in the present case study indicated that the inclusion of turbine-induced turbulence is necessary for accurately predicting near-surface wind speed, near-surface temperature, and wind direction. Also, reducing the turbulence generation in Fitch’s parameterization to 25% of its original value led to an underestimation of near-surface wind speed. The simulations also showed the high impact of interarray wake effects on power generation. For instance, the wake shed by the Veja Mate wind farm reduced the capacity factor of the Bard Offshore 1 wind farm from 0.96 to 0.82 during the same time period under the stable atmospheric conditions of the current case study. These interarray wake effects will be of high importance for future wind farms as the distances between wind farms will become less due to the limited regions of suitable water depths for fixed mounting, while the rated power and occupied surface area of individual wind farms will also be increasing. Even with floating offshore wind turbines, which will enable wind-farm construction in deeper waters, the size and height of modern-day and future wind turbines will enlarge the impact zones of offshore wind farms (e.g., beyond 100 km). Hence, a full understanding of the capabilities and prediction accuracy of alternative wind-farm parameterizations within mesoscale models will be of critical importance and is worth further research.
Acknowledgments.
This research was partly supported by the Supergen Offshore Renewable Energy hub, funded by the Engineering and Physical Science Research Countil (EPSRC) Grant EP/S000747/1. The authors appreciate the guidance provided by Dr. Ralph Burton from the University of Leeds and the constructive comments from the article reviewers. Also, the authors would like to acknowledge the assistance given by Research IT and the use of the Computational Shared Facility at The University of Manchester. This study was funded by the School of Engineering at the University of Manchester.
Data availability statement.
All the input files to WRF to regenerate the experiments in this study, along with the modified WRF modules and relevant files from the datasets to which results were compared, are publicly available in Ali et al. (2022). The figures in this study can be re-generated using the Python code publicly available in https://github.com/Karim-Ali-UOM/WRF-post-processing. The satellite image of the North Sea was downloaded from DTU (2022). The measurements of the FINO-1 meteorological mast can be accessed through http://fino.bsh.de. The airborne measurements over the Gode wind farms are available in Bärfuss et al. (2019).
APPENDIX A
Historical Evolution of Wind-Farm Parameterizations
Here, we chronologically follow the evolution of wind-farm parameterizations that represent wind turbines as sources of turbulence and sinks of momentum. Studies that represent wind turbines as increased surface roughness length can be found in the reviews of Porté-Agel et al. (2020) and Fischereit et al. (2022). Table A1 summarizes some of the details in this appendix.
Summary of wind-farm parameterizations. The abbreviation “NA” refers to “not applicable.”
The formulations of turbine-induced thrust and turbulence have changed substantially in the last two decades. An early transition from the increased surface roughness representation of wind turbines was performed by Keith et al. (2004) in a study considering the global impact of wind-power generation on near-surface temperature. To represent a wind turbine, they added an explicit drag source to the first two layers above the surface in a mesoscale model, regardless of the vertical extent of the wind turbines. The drag source was assumed to be proportional to the resolved wind speed with a proportionality constant that yielded a 40% extraction of resolved kinetic energy per turbine-containing grid cell, whereas turbine-induced turbulence was ignored. However, Baidya Roy et al. (2004) showed that neglecting turbine-induced turbulence can substantially alter the vertical profiles of temperature, humidity, and surface heat fluxes, as well as a mean 28% overestimation of power generation within their considered scenario. Rather, they assumed that wind turbines approximately added a constant amount of TKE to the flow, up to 5 m2 s−2 per wind turbine depending on the amount of resolved kinetic energy in a grid cell. To account for turbine-induced thrust, 40% of a grid cell’s resolved kinetic energy was removed whenever the wind speed in this grid cell exceeded 1 m s−1. However, in a subsequent study (Baidya Roy 2011), the amount of removed kinetic energy was allowed to vary with wind speed within 4–25 m s−1 according to the wind turbine’s performance curve, whereas the amount of added turbulence was kept the same. In these two studies (Baidya Roy et al. 2004; Baidya Roy 2011), turbine rotors were placed within a single grid layer.
In contrast, Blahak et al. (2010) hypothesized that the added turbulence to a grid cell was approximately 20% of the decrease in its resolved kinetic energy due to the existence of a wind turbine rather than adding 5 m2 s−2 to the flow’s TKE per wind turbine. They also formulated their parameterization so that a wind turbine was vertically resolved by multiple grid layers instead of using a single grid layer (Baidya Roy et al. 2004; Baidya Roy 2011) or two grid layers (Keith et al. 2004). For each of these grid layers, a drag source proportional to the wind turbine’s power coefficient (CP), the square of wind speed, and the fraction of the rotor area confined within the grid layer was added. The results of this study were not compared to previous studies nor to LES. However, Abkar and Porté-Agel (2015) compared the parameterization of Blahak et al. (2010) to LES and showed that it underestimated turbine-induced thrust and turbulence.
Fitch et al. (2012) argued that turbine-induced turbulence generation was not constant (Baidya Roy et al. 2004; Baidya Roy 2011) nor was it a fraction of the turbine-induced thrust (Blahak et al. 2010). Rather, they suggested that turbine-induced turbulence is proportional to the difference between the thrust and power coefficients [Eq. (B8)]. Their assumption was based on the premise that the thrust coefficient indicates how much energy is extracted from the flow, whereas the power coefficient represents the amount of electrical energy produced by a wind turbine. Hence, their difference is converted into turbulence if all other mechanical losses are neglected. Nevertheless, they used a formulation for the turbine-induced thrust that was close to that of Blahak et al. (2010), but with the usage of the thrust coefficient (CT) instead of the power coefficient (CP). Being the only wind-farm parameterization publicly available in WRF, multiple studies have tested the performance of Fitch’s parameterization against LES (e.g., Abkar and Porté-Agel 2015; Eriksson et al. 2015; Chatterjee et al. 2016; Vanderwende et al. 2016; Pan and Archer 2018; Peña et al. 2022) and field measurements (e.g., Jiménez et al. 2015; Hasager et al. 2017; Lee and Lundquist 2017; Platis et al. 2018; Siedersleben et al. 2020; Tomaszewski and Lundquist 2020). These studies showed reasonable predictions of turbine-induced thrust. However, some of these studies (e.g., Abkar and Porté-Agel 2015; Eriksson et al. 2015; Siedersleben et al. 2020; Archer et al. 2020) reported that Fitch’s parameterization introduced excessive amounts of turbulence, whereas Archer et al. (2020) suggested that reducing the coefficient of turbulence generation
Rather than using a grid cell’s averaged wind speed to compute the turbine-induced thrust as in Fitch et al. (2012), Boettcher et al. (2015) used a reference simulated wind speed obtained from a reference location that was upwind of a turbine by approximately 10% of its rotor diameter, in an attempt to examine the impact of offshore wind farms on the German Bight. They also neglected turbine-induced turbulence within their parameterization. Their results, however, were not compared to other wind-farm parameterizations nor to LES. Hence, the impact of their assumptions has not been determined. Nevertheless, the assumption of neglecting turbine-induced turbulence is shown in the current study to negatively impact the accuracy of a mesoscale simulation.
Similar to Boettcher et al. (2015), Abkar and Porté-Agel (2015) used a correction factor ζ to account for the difference between the undisturbed wind speed and the volume-averaged wind speed within a turbine-containing grid cell during the calculation of turbine-induced thrust and turbulence. Their formulation of turbine-induced thrust was close to that of Fitch’s parameterization, to which it reverted when the correction factor ζ = 1 [Eqs. (B7) and (B12)]. They also introduced an analytical approximation of turbine-induced turbulence [Eq. (B13)], making it inherently different from that of Fitch’s parameterization, which mainly depended on turbine performance curves. Abkar and Porté-Agel (2015) tested their parameterization and Fitch’s parameterization in LES of a stably stratified boundary layer. Turbine-induced thrust and turbulence were both better predicted using Abkar’s parameterization than Fitch’s parameterization. However, there was no systematic way introduced to assign a value for the correction factor ζ. Rather, Abkar and Porté-Agel (2015) relied on LES of idealized setups to infer a suitable value of ζ, which introduces some uncertainty from not knowing the value of ζ, especially for large domain sizes whose LES is computationally expensive.
To remedy this drawback, Pan and Archer (2018) introduced a regression to compute the correction factor (correction function, hereafter) based on LES. The correction function depended on some geometric parameters of a wind-farm layout such as the percentage of a rotor-plane surface area that is blocked by upwind turbines (Ghaisas and Archer 2016). Within their parameterization, Pan and Archer (2018) used the analytical approximation of turbine-induced turbulence introduced in Abkar’s parameterization, but with ζ = 1. They compared their parameterization to Fitch’s parameterization and to LES by simulating an idealized WRF scenario of the Lillgrund wind farm located in the Øresund, where all the farm’s turbines were placed in a single grid cell. They showed that within this setup, Fitch’s parameterization overestimated turbine-induced turbulence and power generation due to the neglected wake effects within the grid cell containing the wind farm. However, Ma et al. (2022) questioned the generality of the parameterization as the regression was calibrated using the Lillgrund wind farm only.
Fitch’s parameterization and all the subsequent modifications (Boettcher et al. 2015; Abkar and Porté-Agel 2015; Pan and Archer 2018) used hub-height wind speed in thrust and turbulence calculations. However, Redfern et al. (2019) argued that this assumption, though adequate for small wind turbines, may cause inaccuracies in power generation for large wind turbines, because the effects of wind shear and veer (the variation of horizontal wind direction along the vertical extent of a rotor) are neglected. Therefore, they suggested replacing hub-height wind speed with an area-averaged rotor equivalent wind speed [Eq. (B20)]. They also introduced a correction for wind veer effects at each vertical grid level. However, through sets of idealized WRF scenarios, Redfern et al. (2019) showed that the introduced corrections did not cause substantial changes compared to Fitch’s parameterization for stable and neutral conditions. Nevertheless, they suggested that these corrections may have a substantial impact in the cases of low-level jets or near-surface inversions.
The previously mentioned wind-farm parameterizations relied on simplifying Eq. (1) for the calculation of turbine-induced thrust. However, Volker et al. (2015) did not follow the same approach. Rather, they assumed a Gaussian wake distribution for each wind turbine and then calculated the amount of momentum deficit inside a grid cell by spatially integrating the wake profile. They also derived an analytical expression for the turbine-induced turbulence, which was assumed to be negligible in comparison to turbulence production due to wind shear. Simulated wind speed was compared to that of Fitch’s parameterization and to some meteorological mast measurements over the North Sea, indicating a close agreement with measurements for both parameterizations. Moreover, TKE of both parameterizations was compared, but with no reference to determine their accuracy. Nonetheless, Volker’s parameterization produced considerably lower TKE than Fitch’s parameterization.
A common assumption among currently published wind-farm parameterizations is neglecting subgrid wake effects. That is, all wind turbines in the same grid cell are assumed to have the same onset wind speed defined by the cell-averaged wind speed. The exception to this is Pan’s parameterization whose correction function takes subgrid wake effects into account. Because the horizontal grid size defines the limit of the neglected subgrid wake effects, published wind-farm parameterizations suffer from a horizontal grid-size dependency. To remedy this dependency, Ma et al. (2022) added a correction to Fitch’s parameterization based on the top-hat profile of the Jensen wake model (Jensen 1983). Their results, when compared to Fitch’s parameterization, showed that the latter predicted larger power generation due to the neglected subgrid wake effects. However, this parameterization is not included in the current study, because it adopted a different wind turbine representation relying on a blend between Fitch’s parameterization and analytical wake models.
APPENDIX B
Turbine-Induced Thrust and Turbulence Equations
a. Fitch’s wind-farm parameterization
b. Volker’s wind-farm parameterization
c. Abkar’s wind-farm parameterization
d. Pan’s wind-farm parameterization
e. Redfern’s wind-farm parameterization
APPENDIX C
Differences in Turbine-Induced Turbulence
Turbine-induced turbulence within the wind-farm parameterizations mentioned in appendix A was either neglected (Keith et al. 2004; Baidya Roy et al. 2004; Baidya Roy 2011; Boettcher et al. 2015; Volker et al. 2015), dependent on wind turbine’s performance curves (Fitch et al. 2012; Redfern et al. 2019), or an analytical approximation (Abkar and Porté-Agel 2015; Pan and Archer 2018). Here, we review the differences between the last two categories.
The formulation of turbine-induced turbulence [Eq. (B6)] is a balance between a generation coefficient [
(a) Variation of turbulence generation coefficient (
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
(a) Variation of turbulence generation coefficient (
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
(a) Variation of turbulence generation coefficient (
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
Figure C1a shows that
APPENDIX D
The Role of Energy Correction
A schematic of the intersection of two grid levels with a turbine’s rotor plane.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
A schematic of the intersection of two grid levels with a turbine’s rotor plane.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
A schematic of the intersection of two grid levels with a turbine’s rotor plane.
Citation: Monthly Weather Review 151, 9; 10.1175/MWR-D-23-0006.1
APPENDIX E
Difference between TKE and Measured Wind Speed Deviation
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